A solid figure made by joining several solid figures or hollowing out the inside is called a composite solid. Let us find its volume and surface area.
Key Points about Composite Solids
- Find the volume by using “larger solid − hollowed-out solid” or “the sum of the smaller solids”
- Count only the surfaces visible from the outside (including the inner surfaces of hollowed-out parts)
- Subtract the surfaces where solids are joined because they cannot be seen from the outside
Example 1 (A Cone on Top of a Cylinder): Find the volume of a composite solid consisting of a cylinder with a base radius of 3 cm and a height of 8 cm, with a cone of height 4 cm on top. (π=3.14)
Solution:
① Cylinder part = 9 × 3.14 × 8 = 226.08 cm³
② Cone part = (1/3) × 9 × 3.14 × 4 = 37.68 cm³
③ Total = 226.08 + 37.68 = 263.76 cm³
Solution:
① Cylinder part = 9 × 3.14 × 8 = 226.08 cm³
② Cone part = (1/3) × 9 × 3.14 × 4 = 37.68 cm³
③ Total = 226.08 + 37.68 = 263.76 cm³
Example 2 (A Solid Cube with a Cylindrical Hole): A cylindrical hole with a radius of 2 cm and a height of 10 cm is drilled through the center of a cube with side length 10 cm. Find the remaining volume. (π=3.14)
Solution:
① Volume of the cube = 10³ = 1000 cm³
② Volume of the removed cylinder = 4 × 3.14 × 10 = 125.6 cm³
③ Remaining volume = 1000 − 125.6 = 874.4 cm³
Solution:
① Volume of the cube = 10³ = 1000 cm³
② Volume of the removed cylinder = 4 × 3.14 × 10 = 125.6 cm³
③ Remaining volume = 1000 − 125.6 = 874.4 cm³